Variants on Frobenius Intersection Flatness and Applications to Tate Algebras
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arXiv
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| Autores principales: | , , , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866908603993030656 |
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| author | Datta, Rankeya Epstein, Neil Schwede, Karl Tucker, Kevin |
| author_facet | Datta, Rankeya Epstein, Neil Schwede, Karl Tucker, Kevin |
| contents | The theory of singularities defined by Frobenius has been extensively developed for $F$-finite rings and for rings that are essentially of finite type over excellent local rings. However, important classes of non-local excellent rings, such as Tate algebras and their quotients (affinoid algebras) do not fit into either setting. We investigate here a framework for moving beyond the $F$-finite setting, developing the theory of three related classes of regular rings defined by properties of Frobenius. In increasing order of strength, these are Frobenius Ohm-Rush (FOR), Frobenius intersection flat, and Frobenius Ohm-Rush trace (FORT).
We show that Tate algebras are Frobenius intersection flat, from which it follows that reduced affinoid algebras have test elements using a result of Sharp. We also deduce new cases of the openness of the $F$-pure locus. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_06444 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Variants on Frobenius Intersection Flatness and Applications to Tate Algebras Datta, Rankeya Epstein, Neil Schwede, Karl Tucker, Kevin Commutative Algebra Algebraic Geometry 13A35, 13F40, 12J25, 14G22 The theory of singularities defined by Frobenius has been extensively developed for $F$-finite rings and for rings that are essentially of finite type over excellent local rings. However, important classes of non-local excellent rings, such as Tate algebras and their quotients (affinoid algebras) do not fit into either setting. We investigate here a framework for moving beyond the $F$-finite setting, developing the theory of three related classes of regular rings defined by properties of Frobenius. In increasing order of strength, these are Frobenius Ohm-Rush (FOR), Frobenius intersection flat, and Frobenius Ohm-Rush trace (FORT). We show that Tate algebras are Frobenius intersection flat, from which it follows that reduced affinoid algebras have test elements using a result of Sharp. We also deduce new cases of the openness of the $F$-pure locus. |
| title | Variants on Frobenius Intersection Flatness and Applications to Tate Algebras |
| topic | Commutative Algebra Algebraic Geometry 13A35, 13F40, 12J25, 14G22 |
| url | https://arxiv.org/abs/2504.06444 |